Estimates for a class of oscillatory integrals and decay ...arnold/papers/wave.pdf · equalities,...

Estimates for a class of oscillatory integrals anddecay rates for wave-type equations

Anton Arnold, JinMyong Kim and Xiaohua Yao

Abstract. This paper investigates higher order wave-type equations of theform ∂ttu + P (Dx)u = 0, where the symbol P (ξ) is a real, non-degenerateelliptic polynomial of the order m ≥ 4 on R

n. Using methods from harmonicanalysis, we first establish global pointwise time-space estimates for a class ofoscillatory integrals that appear as the fundamental solutions to the Cauchyproblem of such wave equations. These estimates are then used to establish(pointwise-in-time) Lp

− Lq estimates on the wave solution in terms of theinitial conditions.

Mathematics Subject Classification (2000). 42B20; 42B37; 35L25; 35B65.

Keywords. Oscillatory integral, higher-order wave equation, fundamental so-lution estimate.

1. Introduction

It is well known that the solution u(t, x) of the Cauchy problem for the wave

equation:

∂ttu(t, x)−u(t, x) = 0, (t, x) ∈ R×Rn

u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn

has the following form:

u(t, x) = F−1 cos(|ξ|t)Fu0 + F−1 sin(|ξ|t)|ξ| Fu1, (1.1)

This work was supported by the Postdoctoral Science Foundation of Huazhong University ofScience and Technology in China and the Eurasia-Pacific Uninet scholarship for post-docs in

Austria. The first author was supported by the FWF (project I 395-N16). The third author wassupported by NSFC (No. 10801057), the Key Project of Chinese Ministry of Education (No.109117), NCET-10-0431, and CCNU Project (No. CCNU09A02015).

2 Anton Arnold, JinMyong Kim and Xiaohua Yao

where F (resp. F−1) denotes the Fourier transform (resp. its inverse). On the

other hand,

u(t, x) = F−1 cos(

[1 + |ξ|2]1/2t)

Fu0 + F−1 sin(

[1 + |ξ|2]1/2t)

(1 + |ξ|2)1/2 Fu1 (1.2)

is the solution of the linear Klein-Gordon equation:

∂ttu(t, x)−u(t, x) + u(t, x) = 0, (t, x) ∈ R×Rn

u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn.

If we use P (ξ) = |ξ|2 and P = 1 + |ξ|2, respectively in (1.1) and (1.2), then the

above solutions read as follows:

u(t, x) = F−1 cos(

P 1/2(ξ)t)

Fu0 + F−1 sin(

P 1/2(ξ)t)

P 1/2(ξ)Fu1

=(

F−1 eiP 1/2(ξ)t + e−iP

1/2(ξ)t

2

)

∗ u0 +(

F−1 eiP 1/2(ξ)t − e−iP

1/2(ξ)t

2iP 1/2(ξ)

)

∗ u1.(1.3)

The main focus of this paper is to derive pointwise estimates (both in t and x) on

the oscillatory integrals

I1(t, x) :=

∫

Rn

ei<x,ξ>±itP 1/2(ξ)dξ (1.4)

and

I2(t, x) :=

∫

Rn

ei<x,ξ>±itP 1/2(ξ)P−1/2(ξ)dξ (1.5)

appearing in (1.3) – but for a larger class of symbols P (ξ). From such estimates on

the fundamental solution one can then derive solution properties, like its spatial

decay at a fixed time, or decay/growth estimates of ‖u(t, .)‖Lq in time.

For the classical wave and the Klein-Gordon equations, such pointwise-in-

time Lp–Lq decay estimates (i.e. estimates on ‖u(t, .)‖Lq in terms of ‖u0‖Lp and

‖u1‖Lp) can be found frequently in the literature [8, 29, 25, 26, 27, 39]. It is also

well known that such Lp–Lp′

estimates allow to deduce the famous Strichartz in-

equalities, which are very useful for the analysis of nonlinear wave equations (see

e.g. [16, 19, 38, 36, 40]). More generally, many similar Strichartz-type estimates

(local and global in time, or with certain spatial weights) for second order hy-

perbolic equations have been established in the case of variable coefficients or on

Riemannian manifolds. There, crucial analytic tools from microlocal analysis or

spectral theory are employed (see e.g. [2, 7, 9, 18, 28, 33, 34, 41] and the refer-

ences therein). We remark that these mentioned Strichartz-type estimates are for

space-time-integrals, while our estimates are all pointwise in time.

Oscillatory integrals and decay rates for wave-type equations 3

In this paper, our main aim is to derive Lp–Lq estimates for the following

general wave-type equations:

∂ttu(t, x) + P (−i∇)u(t, x) = 0, (t, x) ∈ R×Rn

u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn ,

where P (ξ) is a positive, real valued polynomial of higher (even) order m ≥ 4 on

Rn. In order to derive Lp–Lq estimates of the solution (1.3), it suffices to study

pointwise estimates of the oscillatory integrals (1.4) and (1.5) associated to the

general polynomial P . To this end, we need the following assumptions on P (ξ):

(H1): P : Rn → R is a real elliptic inhomogeneous polynomial of even order

m ≥ 4 with P (ξ) > 0 for all ξ ∈ Rn, and n ≥ 2.

(H2): P is non-degenerate, i.e. the determinant of the Hessian

det(∂2Pm(ξ)

∂ξi∂ξj

)

n×n6= 0 ∀ ξ ∈ R

n\0 , (1.6)

where Pm is the principal part of P .

It is well known that for elliptic polynomials P , condition (H2) is equivalent

to the following condition (H2′) (see Lemma 2 in [12]).

(H2′): For any fixed z ∈ Sn−1(the unit sphere of Rn), the function ψ(ω) :=

< z, ω > (Pm(ω))−1/m, defined on Sn−1, is non-degenerate at its critical points.

This means: If dωψ, the differential of ψ at a point ω ∈ Sn−1 vanishes, then

d2ωψ, the second order differential of ψ at this point is non-degenerate. Note that

the non-degeneracy of P is also equivalent to det(∂i∂jP (ξ))n×n being an elliptic

polynomial of order n(m− 2).

Particular examples of such higher order wave-type equations have already

been studied in several papers. For P = 1 + |ξ|4 (linear beam equations of forth

order), Levandosky [22] obtained Lp–Lq estimates and space-time integrability

estimates. He used them to study the local existence and the asymptotic behavior

of solutions to the nonlinear equation with nonlinear terms growing like a certain

power of u. Further, Levandosky and Strauss [23], Pausader [30, 31] established

the scattering theory of the nonlinear beam equation with subcritical nonlinear

terms for energy initial values. Even earlier, for P = 1 + |ξ|m (with m ≥ 4 even),

Pecher [32] studied Lp–Lp′

estimates of such higher order wave equations and


also considered their application to nonlinear problems. Clearly, these polynomials

are special cases satisfying our assumptions stated above. In the sequel, we shall

deal with the general class in the form of the oscillatory integrals (1.4) and (1.5)

under the assumptions (H1) and (H2). Comparing with the classical wave equation

and Klein-Gorden equations, the fundamental solutions of higher order wave-type

equations behave “better” in the dispersions relation and w.r.t. the gain of a

certain decay in the space variable x. As a consequence, we can obtain a larger

set of admissible (1/p, 1/q)–pairs such that the Lp–Lq estimates hold (see §4).Concerning dispersive estimates, our methods (mainly from harmonic analysis)

and results are similar to those of various dispersive Schrodinger-type equations.

And on this topic there exists a vast body of literature, see e.g. [1, 3, 4, 5, 6, 10,

11, 12, 14, 15, 21, 20, 26, 42, 43].

The oscillatory integrals (1.4) and (1.5) can initially be understood in the

distributional sense. Based on the assumption that P is elliptic, it is easy to see

that Ij(t, x); j = 1, 2 are infinitely differentiable functions in the x variable for

every fixed t 6= 0 (e.g. see §1 of [12]). In this paper, we shall derive pointwise

time-space estimates for the oscillatory integrals (1.4) and (1.5). Subsequently,

such estimates are used to establish Lp − Lq estimates for the wave solutions.

Finally, we also remark that, based on these Lp−Lq estimates, some applications

to nonlinear problems can be expected, which will be investigated in a following

paper.

This paper is organized as follows. In Section 2, we make some pretreatment

to the oscillatory integrals (1.4) and (1.5), review the (polar coordinate transfor-

mation) method of Balabane et al. [1] and its extension by Cui [12]. In the core

Section 3 we prove the pointwise time-space estimates on (1.4), (1.5), following

the strategy from [21]. Finally, in §4 these estimates are applied to obtain Lp−Lq

estimates for solutions to higher order wave equations.

2. Preliminaries

We denote by Sn−1 the unit sphere in R

n, and by (ρ, ω) ∈ [0,∞)×Sn−1 the polar

coordinates in Rn. Throughout this paper, we assume that P : Rn → R satisfies

the assumptions (H1) and (H2) (or (H2′)). Hence, Pm(ξ) > 0 for ξ 6= 0. This


implies that there exists a large enough constant a > 0 with: For each fixed s ≥ a

and each fixed ω ∈ Sn−1, the equation P (ρω) = s has a unique positive solution

ρ = ρ(s, ω) ∈ C∞([a,∞)× Sn−1). By Lemma 2 in [1], ρ can be decomposed as

ρ(s, ω) = s1m (Pm(ω))−

1m + σ(s, ω), (2.1)

where σ ∈ S01,0([a,∞)×S

n−1). This symbol class denotes functions in C∞([a,∞)×Sn−1) that satisfy the following condition (cf. [12, 37]): For every k ∈ N0 and every

differential operator Lω on the sphere Sn−1, there exists a constant CkL such that

|∂ksLωσ(s, ω)| ≤ CkL(1 + s)−k for s ≥ a and ω ∈ Sn−1. (2.2)

We now recall two lemmata (see [1, 12]) for the following phase function

φ(s, ω) := s−1m ρ(s, ω)〈z, ω〉 for s ≥ a and ω ∈ S

n−1,

with some fixed z ∈ Sn−1. Clearly, φ ∈ S0

1,0([a,∞) × Sn−1). For every fixed

z0 ∈ Sn−1 there exists a (sufficiently small) neighborhood Uz0 ⊂ S

n−1 of z0 such

that the following lemmata hold uniformly in z ∈ Uz0 (i.e. the constants in Lemma

2.1, Lemma 2.2, and Lemma 2.3 are then independent of z). Therefore we do not

write the variable z in the function φ.

Lemma 2.1 (Lemma 4 of [12], Lemma 3 of [1]). There exists a constant a0 ≥ a

and an open cover Ω0,Ω+,Ω- of Sn−1 with Ω+ ∩ Ω- = ∅ such that it holds for

s ≥ a0:

(a) The function Ω0 ∋ ω 7→ φ(s, ω) has no critical points, and

‖dωφ(s, ω)‖ ≥ c > 0 for ω ∈ Ω0, (2.3)

where the constant c is independent of s.

(b) Each of the two functions Ω± ∋ ω 7→ φ(s, ω) has a unique critical point,

which satisfies: ω± = ω±(s) ∈ C∞([a0,∞); Ω′±) for some open subset Ω′

±with

Ω′± ⊂ Ω±, respectively. Furthermore,

‖(d2ωφ(s, ω))−1‖ ≤ c0 for ω ∈ Ω±, (2.4)

where the constant c0 is independent of s. Moreover, lims→∞ ω±(s) exists and

|ω(k)± (s)| ≤ ck(1 + s)−k−

1m for k ∈ N.


Lemma 2.2 (Lemma 6 of [12]). We define φ±(t, r, s) := st+ rs2mφ(s2, ω±(s

2)) for

t, r > 0, and s ≥ a. Then, there exist constants a1 ≥ max(a0,√a) and c2 > c1 > 0

such that we have for s ≥ a1, t > 0, and r > 0:

c1 ≤ ±φ(s, ω±(s)) ≤ c2, (2.5)

∂sφ+(t, r, s) ≥ t+ c1rs2m−1, (2.6)

t− c2rs2m−1 ≤ ∂sφ-(t, r, s) ≤ t− c1rs

2m−1, (2.7)

c1rs2m−2 ≤ |∂2sφ-(t, r, s)| ≤ c2rs

2m−2, (2.8)

and

|∂ksφ±(t, r, s)| ≤ c2rs2m−k for k = 2, 3, · · · . (2.9)

With this preparation we are able to estimate the following oscillatory integral

Φ(λ, s) :=

∫

Sn−1

eiλφ(s,ω)b(s, ω)dω, (2.10)

where b(s, ω) := s1−nm ρn−1∂sρ ∈ S0

1,0([a,∞)×Sn−1) and λ > 0. Let ϕ+, ϕ-, ϕ0 be

a partition of unity of Sn−1, subordinate to the open cover given in Lemma 2.1.

Then we decompose Φ as

Φ(λ, s) = Φ+(λ, s) + Φ-(λ, s) + Ψ0(λ, s),

where

Φ±(λ, s) :=

∫

Sn−1

eiλφ(s,ω)b(s, ω)ϕ±(ω)dω

and

Ψ0(λ, s) :=

∫

Sn−1

eiλφ(s,ω)b(s, ω)ϕ0(ω)dω.

By using the stationary phase method for Ψ0, and Lemma 2.1 and [35] (Corollary

1.1.8, §1.2) for Φ±, one obtains the following result.

Lemma 2.3. For λ > 0 and s > a1 we have

Φ(λ, s) = λ−n−12 eiλφ(s,ω+(s))Ψ+(λ, s) + λ−

n−12 eiλφ(s,ω-(s))Ψ-(λ, s) + Ψ0(λ, s),

(2.11)

where Ψ±, Ψ0 ∈ C∞((0,∞)× [a0,∞)) and

|∂kλ∂jsΨ±(λ, s)| ≤ ck,j(1 + λ)−ks−j for k, j ∈ N0, (2.12)

|∂kλ∂jsΨ0(λ, s)| ≤ ck,j,l(1 + λ)−ls−j for k, j, l ∈ N0. (2.13)


3. Estimates on the oscillatory integrals

In this section we establish pointwise time-space estimates of the oscillatory inte-

grals (1.4) and (1.5). Like in [21], we aim at simultaneous estimates in the time

and spatial variables. This is a refinement of the analysis in [12], where only spatial

decay estimates of the oscillatory integrals are derived. With our refined analysis

we are able to give here global-in-time estimates on the wave solution.

Theorem 3.1. Assume that the polynomial P satisfies the conditions (H1) and (H2)

from §1, and let n ≥ m. Then there exists a constant C > 0 such that

|I2(t, x)| ≤

C|t|−n−m1m1 (1 + |t|− 1

m1 |x|)−µ, for 0 < |t| ≤ 1,

C|t|− 1m (1 + |t|−1|x|)−µ, for |t| ≥ 1,

(3.1)

where m1 := m2 , µ := mn−4n+2m

2(m−2) > 0.

Proof. In the sequel, C denotes some generic (but not necessarily identical) positive

constants, independent of t, ξ, x, and so forth. Since the integrals I2(t, x) and

I2(−t, x) are structurally identical, it suffices to estimate I2(t, x) for t > 0. We

shall now analyze I2 for three different cases of its arguments, starting with the

most delicate situation.

Case (i): t ≥ 1 and r := |x| ≥ t.

Choose ψ ∈ C∞(R) such that

ψ(s) =

0, for s ≤ a11, for s > 2a1,

where a1 is given in Lemma 2.2. We write

I2(t, x) =

∫

Rn

ei(〈x,ξ〉±tP1/2(ξ))P− 1

2 (ξ)ψ(P 1/2(ξ))dξ

+

∫

Rn

ei(〈x,ξ〉±tP1/2(ξ))P− 1

2 (ξ)[1− ψ(P 1/2(ξ))]dξ

=: I21(t, x) + I22(t, x).

First we rewrite I22 as the Fourier transform of a measure, supported on the graph

S := z = ±P 1/2(ξ); ξ ∈ Rn ⊂ R

n+1:

I22(t, x) =

∫

Rn+1

ei(〈x,ξ〉+tz)P− 12 (ξ)[1− ψ(P 1/2(ξ))]δ(z ∓ P (ξ)1/2) dξ dz . (3.2)

Since the polynomial P is of order m, the supporting manifold of the above inte-

grand is of type less or equal m (in the sense of § VIII.3.2, [37]; see the Appendix


in §5). Then, Theorem 2 of § VIII.3 in [37] implies

|I22(t, x)| ≤ C(1 + |t|+ |x|)− 1m ∀t, x. (3.3)

This can be generalized: Since f(t, ξ) := e±itP1/2

P−1/2[1 − ψ(P 1/2)] ∈ C∞c (Rn)

for every t > 0, we obtain by integration by parts

I22(t, x) = i

∫

Rn

ei〈x,ξ〉x

|x|2 · ∇ξf(t, ξ)dξ .

Proceeding recursively this implies (in the spirit of the Paley-Wiener-Schwartz

theorem)

|I22(t, x)| ≤ Cktkr−k for k ∈ N0, x 6= 0, t ≥ 1, (3.4)

and hence also ∀ k ≥ 0. But proceeding as in (3.2) yields the improvement

|I22(t, x)| ≤ Ck|t|−1m (1 + |t|−1|x|)−(k+ 1

m ) for |t| ≥ 1, x ∈ Rn, ∀ k ≥ 0. (3.5)

To estimate I21, we shall derive an ε–uniform estimate of its regularization

Jε(t, x) :=

∫

Rn

e−εP1/2(ξ)+i(〈x,ξ〉±tP 1/2(ξ))P−1/2(ξ)ψ(P 1/2(ξ))dξ for ε > 0.

(3.6)

By the polar coordinate transform and the change of variables (ρ, ω) → (s, ω) such

that ρ = ρ(s, ω) (with P (ρω) = s), we have

Jε(t, x)=

∫ ∞

0

∫

Sn−1

e−εP1/2(ρω)+i(ρ〈x,ω〉±tP 1/2(ρω))P−1/2(ρω)ψ(P 1/2(ρω))ρn−1dωdρ

=

∫ ∞

0

∫

Sn−1

e−ε√s±it√s+irρ〈z,ω〉ψ(

√s)s−1/2ρn−1∂sρdωds (3.7)

=

∫ ∞

0

e−ε√s±it√ss

nm−1s−1/2ψ(

√s)Φ(rs

1m , s)ds

= 2

∫ ∞

0

e−εs±itss2nm −2ψ(s)Φ(rs

2m , s2)ds ,

where z := x/|x| enters in the oscillatory integral Φ from (2.10). For the transfor-

mation ρ→ s we used that ψ(s) = 0 on [0, a1] (see §2 in [12] for a more detailed dis-

cussion). Here and in the sequel we assume that the functions Φ, Φ±, φ±, Ψ±, Ψ0

are smoothly extended to [0, a], in order to write the s–integrals on R+. The pre-

cise form of this extension, however, will not matter – due to the cut-off function

ψ.

The main goal of this proof is to derive, for any z0 ∈ Sn−1, an ε–uniform

estimate of the form |Jε(t, x)| ≤ Ct−νr−µ, with ν := n−mm−2 ≥ 0 (since n ≥ m).

Because of the Lemmata 2.1–2.3, this estimate will hold uniformly on z = x/|x| ∈


Uz0 with a constant C = C(z0). Due to the compactness of Sn−1, finitely many

points z1, ..., zN will suffice to yield a uniform estimate of |Jε(t, x)| on r ≥ t ≥ 1,using C = max

j=1,...,NC(zj). Here, we only consider the case of e−εs+its; for e−εs−its

the estimates are analogous.

Following Lemma 2.3 we decompose Jε as follows:

Jε(t, x) = 2r−n−12

∫ ∞

0

e−εs+iφ+(t,r,s)sn+1m −2ψ(s)Ψ+(rs

2m , s2)ds

+2r−n−12

∫ ∞

0

e−εs+iφ-(t,r,s)sn+1m −2ψ(s)Ψ-(rs

2m , s2)ds

+2

∫ ∞

0

e−εs+itss2nm −2ψ(s)Ψ0(rs

2m , s2)ds

=: R+

ε (t, x) +R-ε(t, x) +R0ε(t, x) ,

where φ± is defined in Lemma 2.2.

We shall first estimate the integralR0ε(t, x) and set v0(s) := s

2nm −2ψ(s)Ψ0(rs

2m , s2).

By the Leibniz rule and (2.13), we have

|v(k)0 (s)| ≤ C(rs2m )−ls

2nm −2−k for l, k ∈ N0,

where r ≥ 1 and s ≥ a1. Choose l ≥ µ ≥ 0 and k ≥ ν ≥ 0. It thus follows by

integration by parts that

|R0ε(t, x)| ≤ Ct−k

∫ ∞

a1

(rs2m )−ls

2nm −2−kds ≤ Ct−kr−l ≤ Ct−νr−µ. (3.8)

To estimate the integral R+

ε (t, x), for given r ≥ t ≥ 1, we set

u+(s) := −εs+ iφ+(t, r, s) ,

v+(s) := sn+1m −2ψ(s)Ψ+(rs

2m , s2)

for s ≥ 0. Since u′+(s) 6= 0 for s ≥ a1, we can define D∗f := (gf)′ for f ∈ C1(0,∞),

where g := −1/u′+. It is not hard to show

Dj∗v+ =

∑

α

cαg(α1) · · · g(αj)v

(αj+1)+ for j ∈ N , (3.9)

where the sum runs over all α = (α1, · · · , αj+1) ∈ Nj+10 such that |α| = j and

0 ≤ α1 ≤ · · · ≤ αj . Since (2.6) and (2.9) imply, respectively, |g(s)| ≤ Cr−1s1−2m

and

|u(k)+ (s)| ≤ Crs2m−k for k = 2, 3, · · · ,

we find by induction on k:

|g(k)(s)| ≤ Cr−1s1−2m−k for k ∈ N0,


which shall yield the spatial decay of I2. To derive the time decay of I2, we note

that (2.6) also implies |g(s)| ≤ t−1. Using this inequality for just one factor in g(k)

we obtain:

|g(k)(s)| ≤ Ct−1s−k for k ∈ N0 .

The novel key step is now to interpolate these two inequalities, which will allow

us to derive estimates also for large time. We have for any θ ∈ [0, 1]:

|g(k)(s)| ≤ Ctθ−1r−θsθ(1−2m )−k for k ∈ N0. (3.10)

On the other hand we have by the Leibniz rule and (2.12):

|v(k)+ (s)| ≤ Csn+1m −2−k for k ∈ N0. (3.11)

It thus follows from (3.9) – (3.11) that

|Dj∗v+(s)| ≤ Ctj(θ−1)r−jθsjθ(1−

2m )+n+1

m −2−j for j ∈ N0, (3.12)

where D0∗v+ := v+. The particular choice θ = µ

n , j = n yields

|Dn∗ v+(s)| ≤ Ctµ−nr−µs

−mn−2n+22m −1. (3.13)

Noting that µ− n < −ν, one gets by integration by parts

|R+

ε (t, x)| = 2r−n−12

∣

∣

∣

∫ ∞

0

eu+(Dn∗ v+)ds

∣

∣

∣≤ Ctµ−nr−

n−12 −µ ≤ Ct−νr−µ.

We now turn to the integral R-ε(t, x). Here we put

u-(s) := −εs+ iφ-(t, r, s) ,

v-(s) := sn+1m −2ψ(s)Ψ-(rs

2m , s2)

for s ≥ 0. We shall denote s0 := (r/t)m

m−2 , c′1 := (c1/2)m

m−2 , and c′2 := (2c2)m

m−2 ,

with c1 and c2 given in Lemma 2.2. Now we decompose R-ε as

R-ε(t, x) = 2r−n−12

∫ c′1s0

0

+

∫ c′2s0

c′1s0

+

∫ ∞

c′2s0

eu-(s)v-(s)ds

=: R-ε1(t, x) +R-ε2(t, x) +R-ε3(t, x) .

This decomposition is motivated by the fact that the phase ∂sφ-(t, r, ·) is negativeon [0, c′1s0), positive on [c′2s0,∞), and is has exactly one zero on [c′1s0, c

′2s0] (cf.

(2.7), (2.8)).

Integrating by parts we obtain

R-ε3(t, x) = 2r−n−12

(eu-(c′

2s0)

u′-(c′2s0)

n−1∑

j=0

(Dj∗v-)(c

′2s0) +

∫ ∞

c′2s0

eu-(Dn∗ v-)ds

)

.


Here and in the sequel, the differential operator D∗f = (gf)′ is considered with

g = −1/u′-. Since (2.7) implies |u′-(s)| ≥ c2rs2m−1 for s ≥ c′2s0, we find that v-(s)

also satisfies (the analogues of) (3.12) and (3.13) for s ≥ c′2s0. If c′2s0 ≤ a1, then

(Dj∗v-)(c′2s0) = 0 for j = 0, · · · , n− 1 (note that ψ ≡ 0 on [0, a1]). Integration by

parts then yields

|R-ε3(t, x)| =∣

∣2r−n−12

∫ ∞

a1

eu-(Dn∗ v-)ds

∣

∣ ≤ Ct−νr−µ,

exactly as done for R+

ε (t, x). If c′2s0 > a1, then

|R-ε3(t, x)| ≤ Cr−n−12

(

(rs2m−10 )−1

n−1∑

j=0

r−js− 2j−n−1

m −20 +

∫ ∞

c′2s0

r−ns−n+m−1

m −1ds)

≤ Cr−n−12 (r−1s

n−m−1m

0

n−1∑

j=0

(rs2m0 )−j + r−ns

−n+m−1m

0 ).

Noting that r ≥ 1, s0 > a1/c′2, and t ≥ 1, it follows that

|R-ε3(t, x)| ≤ Cr−n+12 s

n−m−1m

0 = Ct−νr−µs− 1

20 t−

12 ≤ Ct−νr−µ.

Next we turn to R-ε1(t, x), which is 0 for c′1s0 < a1. If c′1s0 ≥ a1, we use

|u′-(s)| ≥ 12c1rs

2m−1 for a1 ≤ s ≤ c′1s0. Then, a slight modification of the above

method yields again R-ε1(t, x) ≤ Ct−νr−µ.

To estimate R-ε2(t, x), it suffices to estimate the integral

R-0 2(t, x) = 2r−n−12

∫ c′2s0

c′1s0

eiφ-(t,r,s)v-(s)ds

= 2r−n−12 s0

∫ c′2

c′1

eiφ-(t,r,s0τ)v-(s0τ)dτ ,

where the interval of integration is now independent of the parameters t, r. We

obtain from (2.8) that

|∂2τφ-(t, r, s0τ)| ≥ c1rs20(s0τ)

2m−2 ≥ Crs

2m0

for τ ∈ [c′1, c′2]. Since v-(s) also satisfies (the analogue of) (3.11), we obtain by

using (a corollary of) the Van der Corput lemma (cf. [37], p. 334) (uniformly for


ε > 0 small enough):

|R-0 2(t, x)| ≤ Cr−n−12 s0(rs

2m0 )−

12

(

|v-(c′2s0)|+∫ c′2

c′1

|s0v′-(s0τ)|dτ)

≤ Cr−n−12 s0(rs

2m0 )−

12 s

n+1m −2

0

= C t−νr−µ.

The dominated convergence theorem implies that Jε(t, ·) converges (as ε→ 0)

uniformly for x in compact subsets of x ∈ Rn; |x| ≥ 1. By summarizing the

above estimates we have

|I21(t, x)| ≤ Ct−ν |x|−µ for |x| ≥ t ≥ 1,

and hence

|I21(t, x)| ≤ Ct−n2 (1 + t−1|x|)−µ ≤ Ct−

1m (1 + t−1|x|)−µ for |x| ≥ t ≥ 1.

Combining this with the estimate (3.5) on I22 (put k = µ− 1m ), we have

|I2(t, x)| ≤ Ct−1m (1 + t−1|x|)−µ for |x| ≥ t ≥ 1.

Case (ii): t ≥ 1 and |x| ≤ t.

For I21 we shall prove now that

|I21(t, x)| ≤ C|t|−n/2 for |t| ≥ 1 and |x| ≤ |t|. (3.14)

We proceed as in [21] and write the integral I21(t, x) as follows:

I21(t, x) =

∫

Rn

eit(±P1/2(ξ)+〈x/t,ξ〉)P−1/2(ξ)ψ(P 1/2(ξ))dξ

=:

∫

Rn

eitΦ(ξ,x,t)P−1/2(ξ)ψ(P 1/2(ξ))dξ,

but we shall focus on the case Φ = P 1/2(ξ) + 〈x/t, ξ〉, and the other case is

analogous.

Since |x/t| ≤ 1, P 1/2(ξ) ≤ c1|ξ|m1 , and |∇P (ξ)| ≥ c2|ξ|m−1 for large |ξ|, thepossible critical points satisfying

∇ξΦ(ξ, x, t) =∇P (ξ)2P 1/2(ξ)

+x

t= 0

must be located in some bounded ball. In order to apply later the stationary

phase principle, let Ω ⊂ Rn be some open set such that suppψ(P 1/2) ⊂ Ω and

|∇P 1/2(ξ)| ≥ c|ξ|m1−1 on Ω. Note that the constant a1 (from the definition of ψ


and Lemma 2.2) could be increased, if necessary, such that both of those conditions

can hold. Then we decompose Ω into Ω1 ∪ Ω2, where

Ω1 =

ξ ∈ Ω ; |∇P 1/2(ξ) +x

t| < 1

2|∇P 1/2(ξ)|+ 1

and

Ω2 =

ξ ∈ Ω ; |∇P 1/2(ξ) +x

t| > 1

4|∇P 1/2(ξ)|

.

Since |xt | ≤ 1 and |∇P 1/2(ξ)| → ∞ as |ξ| → ∞, Ω1 must be a bounded domain and

includes all critical points of Φ inside Ω. Now we choose smooth functions η1(ξ)

and η2(ξ) such that supp ηj ⊂ Ωj and η1(ξ) + η2(ξ) = 1 in Ω. And we decompose

I21 as

I21(t, x) = I211(t, x) + I212(t, x),

I21j(t, x) :=

∫

Rn

eitΦ(ξ,x,t)ηj(ξ)P−1/2(ξ)ψ(P 1/2(ξ))dξ; j = 1, 2.

To estimate I211 we note that

det(∂ξi∂ξjΦ)n×n(ξ, x, t) = det(∂ξi∂ξjP1/2)n×n(ξ) .

Lemma 5.3 (see the Appendix below) implies that the r.h.s. is nonzero on Ω (if

necessary, we can increase the value of a1 to satisfy the requirement), that is,

the Hessian matrix is non-degenerate on Ω. Moreover, |∂αξ Φ| ≤ Cα on Ω1 for any

multi-index α ∈ Nn0 . Hence we obtain by the stationary phase principle that

|I211(t, x)| ≤ C|t|−n/2. (3.15)

To estimate I212, we shall use some cut-off in order to make the subsequent

integrations by parts meaningful (cp. to the procedure in (3.6)). Using a smooth,

compactly supported cut-off function 0 ≤ ϕ ≤ 1 with ϕ(0) = 1, we shall derive an

ε–uniform estimate (as ε→ 0) of

Iε212(t, x) :=

∫

Rn

eitΦ(ξ,x,t)η2(ξ)ϕ(εξ)P−1/2(ξ)ψ(P 1/2(ξ))dξ .

Note that |∇ξΦ| = |∇P 1/2(ξ) + xt | ≥ 1

4 |∇P 1/2(ξ)| ≥ c|ξ|m1−1 for ξ ∈ Ω2 and

|∂αξ Φ| ≤ Cα|ξ|m1−α for |α| ≥ 2. Now we define the operator L by

Lf :=〈∇ξΦ,∇ξ〉it|∇ξΦ|2

f.


Since LeitΦ = eitΦ, we obtain by N iterated integrations by parts:

|Iε212(t, x)| =

∣

∣

∣

∣

∫

Rn

eitΦ(ξ,x,t)(L∗)N[

ϕ(εξ)η2(ξ)P−1/2(ξ)ψ(P 1/2(ξ))

]

dξ

∣

∣

∣

∣

≤ CN |t|−N∫

suppψ(P 1/2)

|ξ|−mNdξ ≤ C ′N |t|−N , (3.16)

where N > n and L∗ is the adjoint operator of L. Combining the estimates (3.15)

and (3.16) yields the claimed estimate |I21| ≤ C|t|−n/2 for |t| ≥ 1 and |x| ≤ |t|.

Together with the estimate (3.5) (with k = µ− 1m ) on I22 this yields

|I2(t, x)| ≤ Ct−1m (1 + t−1|x|)−µ for t ≥ 1 and |x| ≤ |t|. (3.17)

Thus, combining the cases (i) and (ii), we conclude

|I2(t, x)| ≤ Ct−1m (1 + t−1|x|)−µ for t ≥ 1 and x ∈ R

n. (3.18)

Case (iii): For 0 < t < 1 and x ∈ Rn we shall use a standard scaling argument.

We observe that

I2(t, x) =

∫

Rn

ei(〈x,ξ〉+tP1/2(ξ))P−1/2(ξ)dξ

= t−n

m1

∫

Rn

ei(

〈t−1

m1 x,ξ〉+tP 1/2(t−

1m1 ξ)

)

P−1/2(t−1/m1ξ)dξ

= t−n

m1+1

∫

Rn

ei(

〈t−1

m1 x,ξ〉+(t2P ((t2)−1m ξ))1/2

)

(t2P ((t2)−1m ξ))−1/2dξ.

Let Pt(ξ) := t2P (t−2m ξ), ρt(s, ω) := t

2m ρ( st2 , ω), and σt(s, ω) := t

2mσ( st2 , ω). Then

(2.1) still holds when P , ρ, σ are replaced, respectively, by Pt, ρt, σt. It is easy

to check that σt also satisfies (2.2) with the same constants CkL. Hence, we can

deduce from (3.18) (with t = 1) that

|I2(t, x)| ≤ Ct−n−m1m1 (1 + t−

1m1 |x|)−µ, for t ∈ (0, 1) and x ∈ R

n. (3.19)

This completes the proof of the theorem.

Remark 3.2. If one checks the details of the proof for the cases (i) and (ii) above,

one finds that the estimate for I2(t = 1, x) does not use the condition n ≥ m.

Therefore the estimate (3.19) of I2(t, x) for 0 < t < 1 is also obtained by scaling

without the restriction n ≥ m.


Similarly to the above proof of I2, we obtain the following result for the

oscillatory integral I1(x, t).

Theorem 3.3. Assume that the polynomial P satisfies (H1) and (H2). Then

|I1(t, x)| ≤

C|t|− nm1 (1 + |t|− 1

m1 |x|)−n(m−4)2(m−2) , for 0 < |t| ≤ 1 ,

C|t|− 1m1 (1 + |t|−1|x|)−

n(m−4)2(m−2) , for |t| ≥ 1 .

(3.20)

Note that I1 has the same structure as the oscillatory integral I(t, x) in [21]

for higher order Schrodinger equations, when replacing P 1/2(ξ) from (1.4) by P (ξ).

Thus, Theorem 3.3 is closely related to Theorem 3.1 of [21] (when replacing m1

by m). This similarity is also easily seen on the level of the considered evolution

equations: The differential operator of our wave-type equation can be factored as

∂tt + P (Dx) = [∂t + i√P (Dx)] [∂t − i

√P (Dx)] ,

where each squared bracket corresponds to a time-dependent Schrodinger equation.

4. Decay/growth estimates for wave-type equations

Here we shall apply the Theorems 3.1, 3.3 to establish Lp − Lq estimates for the

solution of the following higher order wave-type equation:

∂ttu(t, x) + P (−i∇)u(t, x) = 0, (t, x) ∈ R×Rn,

u(0, x) = u0(x), ∂tu(0, x) = u1(x), x ∈ Rn.

As in (1.3), its solution is given by

u(t, x) = F−1 cos(

P 1/2(ξ)t)

Fu0 + F−1 sin(

P 1/2(ξ)t)

P 1/2(ξ)Fu1 =: U(t, x) + V (t, x).

For any a ∈ R we define the following set of admissible index pairs.

a := (p, q); ( 1p ,1q ) lies in the closed quadrangle ABCD,

where A = ( 12 ,12 ), B = (1, 1

qa), C = (1, 0), and D = ( 1

q′a, 0) for qa := n

µa, µa :=

mn−4n+2a2(m−2) , and 1

q + 1q′ = 1. Moreover, we denote the Lorentz space by Lp,q(Rn)

(see p.48 in [17]).


1p

1q

12

12 1

A

B

CD

E

F

Solid line: admissible index pairs for Th. 4.1: ( 1p ,1q ) with (p, q) ∈ m. Dashed

line: admissible index pairs for Th. 4.2. Dotted line: admissible index pairs from

Rem. 4.3(2). Figure shows the example m = 4, n = 6.

Theorem 4.1. Assume that the polynomial P satisfies the conditions (H1) and

(H2), and let n ≥ m (and hence 2 ≤ qm <∞). Then we have

‖V (t, ·)‖Lq∗≤

C|t| nm1

( 1q− 1

p )+1‖u1‖Lp∗, for 0 < |t| ≤ 1,

C|t|n|1q− 1

p′|− 1

m ‖u1‖Lp∗, for |t| ≥ 1,

(4.1)

where (p, q) ∈ m, m1 := m/2. Here, the pair of spaces (Lp∗, Lq∗) has the following

meaning:

(Lp∗, Lq∗) =

(L1, Lqm,∞), if (p, q) = (1, qm),

(Lq′

m,1, L∞), if (p, q) = (q′m,∞),(Lp, Lq), otherwise.

(4.2)

Proof. By the assumption (H1), one has∣

∣

∣

∣

∣

sin(

P 1/2(ξ)t)

P 1/2(ξ)

∣

∣

∣

∣

∣

≤

|t|, for 0 < |t| ≤ 1,C, for |t| ≥ 1.

Then, the Plancherel theorem gives the result for the index point A :

‖V (t, ·)‖L2 ≤

|t|‖u1‖L2 , for 0 < |t| ≤ 1,C‖u1‖L2 , for |t| ≥ 1.

(4.3)

On the other hand, by Theorem 3.1 we have for each t 6= 0: I2(t, ·) ∈ Lq(Rn)

∀ q > qm and I2(t, ·) ∈ Lqm,∞(Rn) (the weak Lqm space). Applying the (weak)


Young inequality (see p.22 in [17]) to the second term of (1.3) then implies

‖V (t, ·)‖Lq∗≤

C|t| nm1

( 1q−1)+1‖u1‖L1 , for 0 < |t| ≤ 1,

C|t|nq − 1m ‖u1‖L1 , for |t| ≥ 1.

(4.4)

This proves the estimate for the points (1, 1q ) on the edge CB. Applying the

Marcinkiewicz interpolation theorem (see p.56 in [17]) to (4.3) and (4.4) proves

(4.1) for the points in the closed triangle ABC. By duality, the estimate for the

triangle ADC follows immediately from the result in the triangle ABC (note that

the adjoint operator of I2 ∗ u1 has the same structure). To include the result for

the index point D, we remark that Lq′

m,1 ⊂ (Lqm,∞)∗ (cf. [13]). This completes

the proof of the theorem.

Next we shall complement this result with a straight forward estimation of

V (t, x) = F−1Q(t, ξ)Fu1 , Q(t, ξ) :=sin

(

P 1/2(ξ)t)

P 1/2(ξ).

To this end we define the index points E = (n+m2n , 12 ), F = ( 12 ,n−m2n ).

Theorem 4.2. Let the polynomial P satisfy (H1), and let n ≥ m. Then we have

‖V (t, ·)‖Lq∗≤

C|t| nm1

( 1q− 1

p )+1‖u1‖Lp , for 0 < |t| ≤ 1,C‖u1‖Lp , for |t| ≥ 1,

(4.5)

where ( 1p ,1q ) lies in the closed triangle AEF and m1 := m/2. Here, we denote

Lq∗ := Lq,∞ if 1q = 1

p − m2n . And L

q∗ := Lq, elsewise.

Proof. By the assumption (H1), we have |Q(t, ξ)| ≤ C|ξ|−m1 . And hence, Q(t, ·) ∈L

2nm ,∞(Rn). Since we assumed u1 ∈ Lp for some 1

2 ≤ 1p ≤ n+m

2n , we have Fu1 ∈ Lp′

.

And the Holder inequality for Lorentz spaces (cf. [17]) implies

Q(t, ξ)Fu1 ∈ Lp,∞ , p :=p

p− 1 + pm2n

.

The Hausdorff-Young inequality for Lorentz spaces (cf. [24]) then yields the result

for the edge EF with 1q = 1

p − m2n :

‖V (t, ·)‖Lq,∞ ≤ C‖u1‖Lp , ∀ t ∈ R .

Applying the Marcinkiewicz interpolation theorem (with (4.3)) concludes the proof.


Remark 4.3. 1. The short time behavior of u in Th. 4.1 and Th. 4.2 coincides

for the indices in AEF ∩ABCD. But for large time, the r.h.s. of (4.5) stays

uniformly bounded, which is not always the case in (4.1).

2. A Marcinkiewicz interpolation between the edges EF and BC (plus a duality

argument for 1q <

1p′ ) allows to extend the decay/growth estimate on u to the

closed hexagon AEBCDF . But since this follows exactly the above strategy,

we do not give details here.

3. Theorem 4.2 actually also holds for n < m, but we skipped the statement for

notational simplicity. If m ∈ (n, 2n) one obtains a decay/growth estimate for

the index pair ( 1p ,1q ) in the closed pentagon described by the five different

endpoints: ( 12 ,12 ), (1,

12 ), (1,

n−m1

n ), (m1

n , 0), (12 , 0). And for m ≥ 2n for the

whole index square 12 ≤ 1

p ≤ 1, 0 ≤ 1q ≤ 1

2 .

Now we turn to the estimate of U :

Theorem 4.4. Assume that the polynomial P satisfies (H1) and (H2). Then we

have

‖U(t, x)‖Lq∗≤

C|t| nm1

( 1q− 1

p )‖u0‖Lp∗, for 0 < |t| ≤ 1,

C|t|n|1q− 1

p′|− 1

m1 ‖u0‖Lp∗, for |t| ≥ 1,

(4.6)

where (p, q) ∈ 0 and (Lp∗, Lq∗) is defined in (4.2) (when replacing qm by q0).

Using Th. 3.3, the proof of Th. 4.4 is very similar to Th. 4.1. So we omit the

details here.

Remark 4.5. Let us briefly compare our results to the literature: While Theorem

2.3 of [32] only yields Lp − Lp′

estimates for the case P = 1 + |ξ|m, our Th. 4.1

provides more general Lp − Lq estimates. Moreover, our result applies to more

general polynomials P .

5. Appendix: The type of a hypersurface

In § VIII.3.2 of [37] the type of a hypersurface S := z = Φ(ξ); ξ ∈ Rn ⊂ R

n+1

is defined as follows: The type m(ξ0) of S at ξ0 is the smallest integer k ≥ 2, such

that the matrix (or tensor)(

∂αΦ(ξ0))

|α|=k does not vanish. Then, the type of S

is m := maxξ0∈Rn

m(ξ0).


Lemma 5.1. Let degP (ξ) = m ≥ 4 and P (ξ) > 0 on Rn. Then, the type of

S := z = P 1/2(ξ) satisfies m ≤ m.

Proof. Assume that there exists a ξ0 ∈ Rn with m(ξ0) > m. Since P 1/2 is smooth

we have in a small neighborhood around ξ0:

P 1/2(ξ) = P 1/2(ξ0) + (ξ − ξ0) · ∇ξP1/2(ξ0) +O

(

|ξ − ξ0|m(ξ0))

.

Hence,

P (ξ) = P (ξ0) + 2(ξ − ξ0) · ∇ξP1/2(ξ0) P

1/2(ξ0) +[

(ξ − ξ0) · ∇ξP1/2(ξ0)

]2

+O(

|ξ − ξ0|m(ξ0))

,

which contradicts degP (ξ) = m with m ≥ 4.

Remark 5.2. 1. If we assume m = 2 in Lemma 5.1, we obtain m = 2.

2. In the example P (ξ) = 1+ 2|ξ|2 + |ξ|4 we have P 1/2(ξ) = 1+ |ξ|2, and hence

m = 2 < m. But in general we can only conclude m ≤ m for m ≥ 4.

Lemma 5.3. Let the polynomial P on Rn satisfy (H1) and (H2). Then

det(∂2P 1/2(ξ)

∂ξi∂ξj

)

n×n∼ c(

ξ

|ξ| ) |ξ|n(m

2 −2) for |ξ| large,

where c is a smooth function on the unit sphere of Rn, bounded away from 0.

Proof. Step 1:

With Pm denoting the principal part of P , we define φ(ξ) := P1/mm (ξ), which is

positive for ξ 6= 0 and homogeneous of degree one. Now we consider its level-1-

hypersurface

Σ := ξ ∈ Rn ; φ(ξ) = 1 ⊂ R

n .

Since Pm = φm is non-degenerate by assumption (H2) (i.e. det (∂i∂jφm) 6= 0 for

ξ 6= 0), Proposition 4.2 from [10] implies that Σ is strictly convex and of type 2.

Applying again Proposition 4.2 (with λ = m/2) implies

det(∂2P

1/2m (ξ)

∂ξi∂ξj

)

n×n6= 0 ∀ ξ ∈ R

n\0 . (5.1)

Step 2:

Now we decompose

P 1/2(ξ) = P 1/2m (ξ) + a(ξ) ,


with a = P−Pm√P+

√Pm

∈ Sm2 −1. Hence,

det(∂2P 1/2(ξ)

∂ξi∂ξj

)

= det(∂2P

1/2m (ξ)

∂ξi∂ξj

)

+Q(ξ) ,

where the first term on the r.h.s. is O(

|ξ|n(m2 −2)

)

for ξ large, and the second term

is of the order O(

|ξ|n(m2 −2)−1

)

. The claim then follows from (5.1).

References

[1] M. Balabane and H. A. Emami-Rad, Lp estimates for Schrodinger evolution equations,Trans. Amer. Math. Soc. 292 (1985), 357-373.

[2] M. Ben-Artzi, Eigenfunction expansion and spacetime estimates for generators in

divergence form, Rev. Math. Phy. 22 (2010), 1209-1240.

[3] M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for fourth order

Schrodinger equations, CRAS Paris 330 (2000), 87-92.

[4] M. Ben-Artzi, H. Koch and J.-C. Saut, Dispersion estimates for third order equations

in two dimensions, Comm. PDE 28 (2003), 1943-1974.

[5] M. Ben-Artzi and J.-C. Saut, Uniform estimates for a class of oscillatory integrals

and applications, Diff. and Int. Eq. 12 (1999), 137-145.

[6] M. Ben-Artzi and F. Treves, Uniform estimates for a class of evolution equations, J.Funct. Anal. 120 (1994), 264-299.

[7] J.M Bouclet and N. Tzvetkov, On global Strichartz estimates for non-trapping metrics,J. Funct. Anal. 254 (2008), 1661-1682.

[8] P. Brenner, On Lp−Lp′ estimates for the wave-equation, Math. Z. 145 (1975), 251-25.

[9] N. Burq, Global Strichartz estimates for nontrapping geometries: About an article by

H. Smith and C. Sogge, Comm. PDE 28 (2003), 1675-1683.

[10] W. Chen, C. Miao, and X. Yao, Dispersive estimates with geometry of finite type,Comm. PDE 37 (2012) 479-510.

[11] S. Cui, Point-wise estimates for a class of oscillatory integrals and related Lp− Lq

estimates, J. Fourier Anal. and Appl. 11 (2005), 441-457.

[12] S. Cui, Point-wise estimates for oscillatory integrals and related Lp− Lq estimates:

Multi-dimensional case, J. Fourier Anal. and Appl. 12 (2006), 605-627.

[13] M. Cwikel, The dual of weak Lp, Annales de l’institut Fourier 25 (1975), 81-126.

[14] Y. Ding and X. Yao, Lp−Lq estimates for dispersive equations and related applica-

tions, J. Math. Anal. and Appl. 356 (2009), 711-728.

[15] Y. Ding and X. Yao, Hp−Hq estimates for dispersive equations and related appli-

cations, J. Funct. Anal. 257 (2009), 2067-2087.

[16] J. Ginibre and G. Velo. Generalized Strichartz inequalities for the wave equation. J.Funct. Anal. 133 (1995), 50-68.

[17] L. Grafakos, Classical and modern Fourier analysis, Prentice Hall, New Jersey, 2003.

[18] L.V. Kapitanskii, The Cauchy problem for a semilinear wave equation, Zap. Nauchn.Sem. Leningrad Otdel. Mat. Inst. Steklov (LOMI) 181-182 (1990) 24-64, and 38-85.


[19] M. Keel and T. Tao, Endpoint Strichart estimates, Amer. J. Math. 120 (1998),360-413.

[20] C.E. Kenig, G. Ponce, and L. Vega, Oscillatory integrals and regularity of dispersive

equations, Indiana Univ. Math. J. 40 (1991), 33-69.

[21] J.M. Kim, A. Arnod, X. Yao, Global estimates of fundamental solutions for higher-

order Schrodinger equations, To appear in Monatsh. Math. 2012.

[22] S. Levandosky, Decay estimates for the fourth-order wave equaitons, J. DifferentialEquations 143 (1998), 360-413.

[23] S. Levandosky and W. Strauss, Time decay for the nonlinear beam equation, Meth-ods and Applications of Analysis 7 (2000), 783-798.

[24] L. Maligranda and L.E. Persson, Inequalities and interpolation, Collect. Math. 44(1993), 181-199.

[25] B. Marshall, W. Strauss and S. Wainger, Lp− Lq estimates for the Klein-Gordon

equation, J. Math. Pure Appl. 59 (1980), 417-440.

[26] C. Miao, Harmonic Analysis and its Application in Partial Differential Equations,(in Chinese) Science Press, Beijing, 2006.

[27] A. Miyachi, On some estimates for the wave equation in Lp and Hp, J. Fac. Sci.Univ. Tokyo, 27 (1980), 331-354.

[28] G. Mockenhaupt, A. Seeger, and C. Sogge. Local smoothing of Fourier integral op-

erators and Carleson-Sjolin estimates, J. Amer. Math. Soc. 6 (1993), 65-130.

[29] C.S. Morawetz, Time decay for the Klein-Gordon equation, Proc. Roy. Soc. A 306(1968), 291-296.

[30] B. Pausader, Scattering and the Levandosky-Strauss conjecture for fourth order non-

linear wave equations, J. Differential Equations 241 (2) (2007), 237-278.

[31] B. Pausader, Scattering for the Beam equation in low dimensions, Indiana Univ.Math. J. 59 (3) (2010), 791-822.

[32] H. Pecher, Lp-Abschatzungen und klassische Losungen fur nicht-lineare Wellengle-

ichungen, Math. Z. 150 (1976), 159-183.

[33] H. Smith and C.D. Sogge, On the critical semilinear wave equation outside convex

obstacles, J. Amer. Math. Soc. 8 (1995), 879-916.

[34] H. Smith and C.D. Sogge, Global Strichartz estimates for nontrapping perturbations

of the Laplacian, Comm. PDE 25 (2000), 2171-2183.

[35] C.D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Univ. Press, Cam-bridge, 1993.

[36] C.D. Sogge, Lecture on Nonlinear Wave Equations, International Press Publications,1995.

[37] E. M. Stein, Harmonic Analysis: Real Variable Method, Orthogonality and Oscilla-

tory Integrals, Princeton Univ. Press, New Jersey, 1993.

[38] W. A. Strauss, Nonlinear wave equations, CBMS Regional Conference Series inMathematics, No.73, 1989.

[39] R. Strichartz, A priori estimates for the wave equations and some applications, J.Funct. Anal. 5 (1970), 218-235.

[40] T. Tao, Nonlinear dispersive equations, local and global analysis, CBMS RegionalConference Series in Mathematics, No.106, 2006.


[41] D. Tataru. Strichartz estimates for second order hyperbolic operators with nonsmooth

coeffcients III. J. Amer. Math. Soc. 15 (2001), 385-423.

[42] X. Yao and Q. Zheng, Oscillatory integrals and Lpestimates for Schrodinger equa-

tions, J. Diff. Eq. 244 (2008), 741-752.

[43] Q. Zheng, X. Yao and D. Fan, Convex hypersurface and Lp estimates for Schrodinger

equations, J. Funct. Anal. 208 (2004), 122-139.

Anton ArnoldInstitut fur Analysis und Scientific Computing,Technische Universitat WienWiedner Hauptstr. 8, A-1040 Wien, Austria;e-mail: [email protected]

JinMyong Kim(Current Address) : Institut fur Analysis und Scientific Computing,Technische Universitat WienWiedner Hauptstr. 8, A-1040 Wien, Austria;

(Permanent Address) : Department of Mathematics,Kim Il Sung UniversityPyongyang, DPR Korea;e-mail: [email protected]

Xiaohua YaoDepartment of MathematicsCentral China Normal UniversityWuhan 430079, P. R. China;e-mail: [email protected]

Estimates for a class of oscillatory integrals and decay ...arnold/papers/wave.pdf · equalities,...

Documents

Transcript of Estimates for a class of oscillatory integrals and decay ...arnold/papers/wave.pdf · equalities,...